Let Ω be a bounded symmetric domain,Γ ∈Aut(Ω) be a torsion-free lattice,X:= Ω/Γ.Let∈ Z X be an irreducible quasi-projective variety such that Z is the Zariski closure of an infinite family of totally geodesic complex subvarieties Sa∈ Z,a∈ A.Under certain non-degeneracy conditions one expects Z to be also totally geodesic,so that Z is in particular again uniformized by a bounded symmetric domain.This set-up is related to the Andr′e-Oort Conjecture since (positive-dimensional) special varieties in the context of the latter conjecture are known to be totally geodesic.